Efficient Inference of
Gaussian-Process-Modulated Renewal Processes
with Application to Medical Event Data
Thomas A. Lasko
Computational Medicine Laboratory, Department of Biomedical Informatics
Vanderbilt University School of Medicine
Nashville, TN 37203
Abstract
The episodic, irregular and asynchronous nature
of medical data render them difficult substrates
for standard machine learning algorithms. We
would like to abstract away this difficulty for
the class of time-stamped categorical variables
(or events) by modeling them as a renewal process and inferring a probability density over nonparametric longitudinal intensity functions that
modulate the process. Several methods exist
for inferring such a density over intensity functions, but either their constraints prevent their
use with our potentially bursty event streams,
or their time complexity renders their use intractable on our long-duration observations of
high-resolution events, or both. In this paper
we present a new efficient and flexible inference method that uses direct numeric integration and smooth interpolation over Gaussian processes. We demonstrate that our direct method is
up to twice as accurate and two orders of magnitude more efficient than the best existing method
(thinning). Importantly, our direct method can
infer intensity functions over the full range of
bursty to memoryless to regular events, which
thinning and many other methods cannot do. Finally, we apply the method to clinical event data
and demonstrate a simple example application
facilitated by the abstraction.
1
INTRODUCTION
One of the hurdles for identifying clinically meaningful
patterns in medical data is the fact that much of that data
is sparsely, irregularly, and asynchronously observed, rendering it a poor substrate for many pattern recognition algorithms.
A large class of this problematic data in medical records
is time-stamped categorical data such as billing codes.
For example, an ICD-9 billing code with categorical label 714.0 (Rheumatoid Arthritis) gets attached to a patient
record every time the patient makes contact with the healthcare system for a problem or activity related to her arthritis.
The activity could be an outpatient doctor visit, a laboratory test, a physical therapy visit, a discharge from an inpatient stay, or any other billable event. These events occur at
times that are generally asynchronous with events for other
conditions.
We would like to learn things from the patterns of these
clinical contact events both within and between diseases,
but their often sparse and irregular nature makes it difficult to apply standard learning algorithms to them. To abstract away this problem, we consider the data as streams
of events, one stream per code or other categorical label.
We model each stream as a modulated renewal process and
use the process’s modulation function as the abstract representation of the label’s activity. The modulation function
provides continuous longitudinal information about the intensity of the patient’s contact with the healthcare system
for a particular problem at any point in time. Our goal is to
infer these functions, the renewal-process parameters, and
the appropriate uncertainties from the raw event data.
We have previously demonstrated the practical utility of using a continuous function density to couple standard learning algorithms to sparse and irregularly observed continuous variables (Lasko et al., 2013). Unfortunately, the
method of inferring such densities for continuous variables is not applicable to categorical variables. This paper
presents a method that achieves the inference for categorical variables.
Our method models the log intensity functions nonparametrically as Gaussian processes, and uses Markov Chain
Monte Carlo (MCMC) to infer a posterior distribution over
intensity functions and model parameters given the events
(Section 2).
There are several existing approaches to making this inference (Section 3), but all of the approaches we found have
either flexibility or scalability problems with our clinical
data. For example, clinical event streams can be bursty, and
some existing methods are unable to adapt to or adequately
represent bursty event patterns.
In this paper we demonstrate using synthetic data that our
approach is up to twice as accurate, up to two orders of
magnitude more efficient, and more flexible than the best
existing method (Section 4.1). We also demonstrate our
method using synthetic data that mimics our clinical data,
under conditions that no existing method that we know of is
able to satisfactorily operate (Section 4.1). Finally, we use
our method to infer continuous abstractions over real clinical data (Section 4.2), and as a simple application example
we infer latent compound diseases from a complex patient
record that closely correspond to its documented clinical
problems.
2
MODULATED RENEWAL PROCESS
EVENT MODEL
A renewal process models random events by assuming that
the interevent intervals are independent and identically distributed (iid). A modulated renewal process model drops
the iid assumption and adds a longitudinal intensity function that modulates the expected event rate with respect to
time.
We consider a set of event times T = {t0 , t1 , . . . , tn } to
form an event stream that can be modeled by a modulated
renewal process. For this work we choose a modulated
gamma process (Berman, 1981), which models the times
T as
P(T ; a, λ(t)) =
n
1 Y
λ(ti )(Λ(ti ) − Λ(ti−1 ))a−1 e−Λ(tn ) , (1)
Γ(a)n i=1
where Γ(·) is the gamma function, a > 0 is the shape parameter, λ(t) > 0 is the modulating intensity function, and
Rt
Λ(t) = 0 λ(u) du.
Equation (1) is a generalization of the homogeneous
gamma process γ(a, b), which models the interevent intervals xi = ti − ti−1 , i = 1 . . . n as positive iid random variables:
1
γ(x|a, b) = P(x; a, b) =
xa−1 e−x/b ,
Γ(a)ba
(2)
where b takes the place of a now-constant 1/λ(t), and can
be thought of as the time scale of event arrivals.
The intuition behind (1) is that the function Λ(t) warps the
event times ti into a new space where their interevent intervals become draws from the homogeneous gamma process
of (2). That is, the warped intervals wi = Λ(ti ) − Λ(ti−1 )
are modeled by wi ∼ γ(a, b).
For our purposes, a gamma process is better than the simpler and more common Poisson process because a gamma
process allows us to model the relationship between neighboring events, instead of assuming them to be independent
or memoryless. Specifically, parameterizing a < 1 models
a bursty process, a > 1 models a more regular or refractory process, and a = 1 produces the memoryless Poisson
process. Clinical event streams can behave anywhere from
highly bursty to highly regular.
We
model
the
log
intensity
function
log λ(t) = f (t) ∼ GP(0, C) as a draw from a Gaussian process prior with zero mean and the squared
exponential covariance function
ti −tj 2
,
(3)
C(ti , tj ) = σe− l
where σ sets the magnitude scale and l sets the time scale of
the Gaussian process. We choose the squared exponential
because of its smoothness guarantees that are relied upon
by our inference algorithm, but other covariance functions
could be used.
In our application the observation period generally starts at
tmin < t0 , and ends at tmax > tn , and no events occur at
these endpoints. Consequently, we must add terms to (1)
to account for these partially observed intervals. For efficiency in inference, we estimate the probabilities of these
intervals by assuming that w0 and wn+1 are drawn from
a homogeneous γ(1, 1) process in the warped space. The
probability of the leading interval w0 R= Λ(t0 ) − Λ(tmin ) is
∞
then approximated by P(w ≥ w0 ) = w0 e−w dw = e−w0 ,
which is equivalent to w0 ∼ γ(1, 1). The trailing interval
is treated similarly.
Our full generative model is as follows:
1. l ∼ Exponential(α)
log σ ∼ Uniform(log σmin , log σmax )
log a ∼ Uniform(log amin , log amax )
b=1
2. f (t) ∼ GP(0, C) using (3)
3. λ(t) = ef (t)
Rt
4. Λ(t) = 0 λ(u) du
5. w0 ∼ γ(1, 1); wi>0 ∼ γ(a, b)
Pi
6. ti = tmin + Λ−1 ( j=0 wj )
Step 1 places a prior on l that prefers smaller values, and
uninformative priors on a and σ. We set b = 1 to avoid
an identifiability problem. (Rao and Teh (2011) set b =
1/a to avoid this problem. While that setting has some
desirable properties, we’ve found that setting b = 1 avoids
more degenerate solutions at inference time.)
2.1
INFERENCE
Given a set of event times T , we use MCMC to simultaneously infer posterior distributions over the intensity func-
tion λ(T ) and the parameters a, σ, and l (Algorithm 1).
For simplicity we denote g(T ) = {g(t) : t ∈ T } for any
function g that operates on event times. On each round
we first use slice sampling with surrogate data (Murray
and Adams, 2010, code publicly available) to compute new
draws of f (t), σ, and l using (1) as the likelihood function
(with additional factors for the incomplete interval at each
end). We then sample the gamma shape parameter a using
Metropolis-Hastings updates.
One challenge of this
R t direct inference is that it requires integrating Λ(t) = 0 λ(u) du, which is difficult because
λ(t) does not have an explicit expression. Under certain
conditions, the integral of a Gaussian process has a closed
form (Rasmussen and Gharamani, 2003), but we know of
no closed form for the integral of a log Gaussian process.
Instead, we compute the integral numerically (the trapezoidal rule works fine), relying on the smoothness guarantees provided by the covariance function (3) to provide high
accuracy.
The efficiency bottleneck of the update is the O(m3 ) complexity of updating the Gaussian process f at m locations,
due to a matrix inversion. Naively, we would compute f at
all n of the observed ti , with additional points as needed for
accuracy of the integral. To improve efficiency, we do not
directly update f at the ti , but instead at k uniformly spaced
−tmin
. We
points T̂ = {t̂j = tmin + jd}, where d = tmaxk−1
then interpolate the values f (T ) from the values of f (T̂ ) as
needed. We set the number of points k by the accuracy required for the integral. This is driven by our estimate of the
smallest likely Gaussian process time scale lmin , at which
point we truncate the prior on l to guarantee d lmin ≤ l.
The efficiency of the resulting update is O(k 3 )+O(n), with
k depending only on the ratio lmin /(tmax − tmin ).
It helps that the factor driving k is the time scale of changes
in the intensity function λ(t) instead of the time scale of interevent intervals, which is usually much smaller. In practice, we’ve found k = 200 to work well for nearly all of our
medical data examples, regardless of the observation time
span, resulting in an update that is linear in the number of
observed points.
Additionally, the regular spacing in T̂ means that its covariance matrix generated by (3) is a symmetric positive
definite Toeplitz matrix, which can be inverted or solved in
a compact representation as fast as O(k log2 k) (Martinsson et al., 2005). We did not include this extra efficiency in
our implementation, however.
3
RELATED WORK
There is a growing literature on finding patterns among
clinical variables such as laboratory tests that have both a
timestamp and a numeric value (Lasko et al., 2013), but
we are not aware of any existing work exploring unsuper-
Algorithm 1 Intensity Function and Parameter Update
Input: Event times T , regular grid T̂ , current function f
and parameters σ, l, and a
Output: Updated f , λ, Λ, σ, l, and a with likelihood p
1: Update f (T̂ ), σ, l, using slice sampling
2: Compute f (T ) by smooth interpolation of f (T̂ )
3: λ(T ∪ T̂ ) ← ef (T ∪T̂ )
4: Compute Λ(T ) from λ(T̂ ) numerically
5: Compute p = P (T ; a, λ(T )) using (1)
6: Update a and p with Metropolis-Hastings and (1)
vised, data-driven abstractions of categorical clinical event
streams that we address here.
There is much prior work on methods similar to ours that
infer intensity functions for modulated renewal processes.
The main distinction between these methods lies in the
way they handle the form and integration of the intensity
function λ(t). Approaches include using kernel density estimation (Ramlau-Hansen, 1983), using parametric intensity functions (Lewis, 1972), using discretized bins within
which the intensity is considered constant (Moller et al.,
1998, Cunningham et al., 2008), or using a form of rejection sampling called thinning (Adams et al., 2009, Rao and
Teh, 2011) that avoids the integration altogether.
The binned time approach is straightforward, but there
is inherent information loss in the piecewise-constant intensity approximation that it must adopt. Moreover,
when a Gaussian process is used to represent this intensity function, the computational complexity of inference is cubic with the number of bins in the interval of observation. For our data, with events at 1-day
or finer time resolution over up to a 15 year observation period, this method is prohibitively inefficient. A
variant of the binned-time approach that uses variablesized bins (Gunawardana et al., 2011, Parikh et al., 2012)
has been applied to medical data (Weiss and Page, 2013).
This variant is very efficient, but is restricted to a Poisson
process (fixed a = 1), and the inferred intensity functions
are neither intended to nor particularly well suited to form
an accurate abstraction over the raw events.
Thinning is a clever method, but it is limited by the requirement of a bounded hazard function, which prevents
it from being used with bursty gamma processes. (Bursty
gamma processes have a hazard function that is unbounded
at zero). One thinning method has also adopted the use
of Gaussian processes to modulate gamma processes (Rao
and Teh, 2011). But in addition to not working with bursty
events, it also rather inefficient; its time complexity is cubic
in the number of events that would occur if the maximum
event intensity were constant over the entire observation
time span. For event streams with a small dynamic range
of intensities, this is not a big issue, but our medical data
4
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Figure 1: Our direct method (top) is more accurate than thinning (bottom) on parametric intensity functions λ1 to λ3 (left
to right). Red line: true normalized intensity function λ(t)/a; White line: mean inferred normalized intensity function;
Blue region: 95% confidence interval. Inset: inferred distribution of the gamma shape parameter a, with the true value
marked in red. Grey bar at a = 1 for reference.
sequences can have a dynamic range of several orders of
magnitude.
Our method therefore has efficiency and flexibility advantages over existing methods, and we will demonstrate in the
experiments that it also has accuracy advantages.
4
EXPERIMENTS
In these experiments, we will refer to our inference
method as the direct method because it uses direct
numerical integration. A full implementation of our
method and code to reproduce the results on the synthetic data is available at https://github.com/
ComputationalMedicineLab/egpmrp.
We tested the ability of the direct method and the thinning
method to recover known intensity functions and shape
parameters from synthetic data. We then used the direct
method to extract latent intensity functions from streams
of clinical events, and we inferred latent compound conditions from the intensity functions for a complex patient
record that accurately correspond to the dominant diseases
documented in the record.
4.1
SYNTHETIC DATA
Our first experiments were with the three parametric intensity functions below, carefully following Adams et al.
(2009) and Rao and Teh (2011). We generated all data using the warping model described in Section 2, with shape
parameter a = 3.
2
1. λ1 (t)/a = 2e−t/15 + e−((t−25)/10) over the interval
[0, 50], 48 events.
2. λ2 (t)/a = 5 sin(t2 ) + 6 on [0, 5], 29 events.
3. λ3 (t)/a is the piecewise linear curve shown in Figure
1, on the interval [0, 100], 230 events.
We express these as normalized intensities λ(t)/a, which
have units of “expected number of events per unit time”,
because they are more interpretable than the raw intensities
and they are comparable to the previous work done using
Poisson processes, where a = 1.
We compared the direct method to thinning on these
datasets, using the MATLAB implementation for thinning
that was used by Rao and Teh (2011). Adams et al. (2009)
compared thinning to the kernel smoothing and binned time
methods (all assuming a Poisson process), and Rao and
Teh (2011) compared thinning to binned time, assuming
a gamma process with constrained a > 1. Both found thinning to be at least as accurate as the other methods in most
tests.
We computed the RMS error of the true vs. the median normalized inferred intensity, the log probability of the data
given the model, and the inference run time under 1000
burn-in and 5000 inference MCMC iterations.
On these datasets the direct method was more accurate than
thinning for the recovery of both the intensity function and
the shape parameter, and more efficient by up to two orders of magnitude (Figure 1 and Table 1). The results for
thinning are consistent with those previously reported (Rao
and Teh, 2011), with the exception that the shape parameter
inference was more accurate in the earlier results.
0.8
0.2
0
0.5
0.15
0.8
Intensity
Intensity
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0 0.5 1
2
0.1
0.05
0
0
0
500
1000
1500
Time
2000
2500
3000
0
500
1000
1500
Time
2000
2500
3000
Figure 2: Accurate recovery of intensity function and parameters under conditions that would be prohibitive for any other
method of which we are aware. Left panel presents results for high intensities and many events, right panel for low
intensities and few events. While there is insufficient evidence in the right panel to recover the true intensity, the inferred
intensity is reasonable given the evidence, and the inferred confidence intervals are accurate in that the true intensity is
about 95% contained within them. Legend as in Figure 1.
The confidence intervals from the direct method are subjectively more accurate than from the thinning method. (That
is, the 95% confidence intervals from the direct method
contain the true function for about 95% of its length in each
case). This is particularly important in the case of small
numbers of events that may not carry sufficient information
for any method to resolve a highly varying function.
As might be expected, we found the results for λ2 (t) to
be sensitive to the prior distribution on l, given the small
amount of evidence available for the inference. Following
Adams et al. (2009) and Rao and Teh (2011), we used a lognormal prior with a mode near l = 0.2, tuned slightly for
each method to achieve the best results. We also allowed
thinning to use a log-normal prior with appropriate modes
for λ1 (t) and λ3 (t), to follow precedent in the previous
work, although it may have conferred a small advantage to
thinning. We used the weaker exponential prior on those
datasets for the direct method.
Our next experiments were on synthetic data generated to
resemble our medical data. We tested several configurations over wide ranges of parameters, including some that
were not amenable to any known existing approach (such
as the combination of a < 1, high dynamic range of inTable 1: Performance on Synthetic Data. RMS: root-meansquared error; LP: log probability of data given the model;
RT: run time in seconds. Best results for each measure are
bolded.
λ1
λ2
λ3
RMS
0.37
3.1
0.25
Direct
LP
+12.1
−228
+1.21
RT
453
511
385
RMS
0.66
3.4
0.53
Thinning
LP
RT
−62.7 4816
−333
1129
−82.2 41291
tensity, and high ratio of observation period to event resolution, Figure 2). The inferred intensities and gamma parameters were consistently accurate. Estimates of the confidence intervals were also accurate, including in cases with
low intensities and few events (Figure 2, right panel).
4.2
CLINICAL DATA
We applied the direct method to sequences of billing codes
representing clinical events. After obtaining IRB approval,
we extracted all ICD-9 codes from five patient records with
the greatest number of such codes in the deidentified mirror of our institution’s Electronic Medical Record. We arranged the codes from each patient record as streams of
events grouped at the top (or chapter) level of the ICD9 disease hierarchy (which collects broadly related conditions), as well as at the level of the individual disease.
For the streams of grouped events, we included an event if
its associated ICD-9 code fell within the range of the given
top-level division. For example, any ICD-9 event with a
code in the range [390 – 459.81] was considered a Cardiovascular event. While intensity functions are only strictly
additive for Poisson processes, we still find the curves of
grouped events to be informative.
We inferred intensity functions for each of these event
streams (for example, Figure 3). Each curve was generated using 2000 burn-in and 2000 inference iterations in
about three minutes using unoptimized MATLAB code on
a single desktop CPU. The results have good clinical face
validity.
There is much underlying structure in these events that can
now be investigated with standard learning methods applied to the inferred intensity functions. As a simple example, singular value decomposition (Strang, 2003) can infer
the latent compound conditions (which we might as well
Intensity
400
Congenital (386)
Cardiovascular (310)
Dermatologic (9)
Endocrine/Metabolic (115)
Eye/Ear (14)
Gastrointestinal (42)
Genitourinary (263)
Hematologic (54)
Infectious (211)
Injury/Toxicity (52)
Musculoskeletal (1)
Neurologic (10)
Obsetrics (0)
Oncology (4)
Perinatal (27)
Psychiatry (4)
Pulmonary (356)
200
0
0
1
2
Time (years)
3
Figure 3: Inferred intensities for all top-level ICD-9 disease divisions of a very complicated patient’s record. Such a display
may be clinically useful for getting a quick, broad understanding of a patient’s medical history, including quickly grasping
which conditions have not been diagnosed or treated. Numbers in parentheses: total number of events in each division. For
clarity, confidence intervals are not shown.
previously reported methods to construct similar curves
from observations with both a time and a continuous value
(Lasko et al., 2013). These two data types represent the
majority of the information in a patient record (if we consider words and concepts in narrative text to be categorical variables), and opens up many possibilities for finding
meaningful patterns in large medical datasets.
Normalized ith singular value
0.4
0.3
0.2
0.1
0
0
5
10
i
15
20
Figure 4: A small number of latent compound conditions
(or eigendiseases) produce most of the activity captured by
the patient record in Figure 3.
call eigendiseases) underlying the recorded clinical activity, taking into account the continuously changing, longitudinal time course of that activity. The singular values for
the curves in Figure 3 reveal that about 40% of the patient’s
activity relates to a single eigendisease, and about 70% is
distributed among the top three (Figure 4). The inferred
eigendiseases closely correspond to the dominant clinical
problems described in the record (Figure 5).
5
DISCUSSION
We have made two contributions with this paper. First, we
presented a direct numeric method to infer a distribution of
continuous intensity functions from a set of episodic, irregular, and discrete events. This direct method has increased
efficiency, flexibility, and accuracy compared to the best
prior method. Second, we presented results using the direct method to infer a continuous function density as an abstraction over episodic clinical events, for the purposes of
transforming the raw event data into a form more amenable
to standard machine learning algorithms.
The clinical interpretation of these intensity functions is
that increased intensity represents increased frequency of
contact with the healthcare system, which usually means
increased instability of that condition. In some cases, it
may also mean increased severity of the condition, but not
always. If a condition acutely increases in severity, this represents an instability and will probably generate a contact
event. On the other hand, if a condition is severe but stably
so, it may not necessarily require high-frequency medical
contact.
The methods described here to represent categorically labeled events in time as continuous curves augment our
The practical motivation for this work is that once we have
the continuous function densities, we can use them as inputs to a learning problem in the time domain (such as identifying trajectories that may be characteristic of a particular
disease), or by aligning many such curves in time and looking for useful patterns in their cross-sections (which to our
knowledge has not yet been reported). We presented a simple demonstration of inferring cross-sectional latent factors
from the intensity curves of a single record. Discovering
similar latent factors underlying a large population is a focus of future work.
We discovered incidentally that a presentation such as Figure 3 appears to be a promising representation for efficiently summarizing a complicated patient’s medical history and communicating that broad summary to a clinician.
The presentation could allow drilling-down to the intensity
plots of the specific component conditions and then to the
raw source data. (The usual method of manually paging
through the often massive chart of a patient to get this information can be a tedious and frustrating process.)
One could also imagine presenting the curves not of the
raw ICD-9 codes, but of the inferred latent factors underlying them, and drilling down into the rich combinations of
test results, medications, narrative text, and discrete billing
events that comprise those latent factors.
We believe that these methods will facilitate Computational
Phenotype Discovery (Lasko et al., 2013), or the datadriven search for population-scale clinical patterns in existing electronic medical records that may illuminate previously unknown disease variants, unanticipated medication
effects, or emerging syndromes and infectious disease outbreaks.
Acknowledgements
This work was funded by grants from the Edward
Mallinckrodt, Jr. Foundation and the National Institutes
of Health 1R21LM011664-01. Clinical data was provided
by the Vanderbilt Synthetic Derivative, which is supported
by institutional funding and by the Vanderbilt CTSA grant
ULTR000445. Thanks to Vinayak Rao for providing the
thinning code.
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